The 9/11 Forum

This thread is to document the results obtained from playing around with a physics engine. A game physics engine. How useful could that be? If it's accurate for the application, how bad can it be? How is accuracy determined?

Questions, questions, questions. Too tiresome, let me skip to the answers and then go back and revisit the questions.


- A body cannot be simultaneously both rigid and deformable
- A rigid body, obeying d'Alembert's principle, forces energy dissipation of crushing to occur in the lower block
- A non-rigid upper block passes diminished jolt amplitude to the 'roofline' but does so by dissipating energy internally, through deformation and failure
- A discrete model, even highly idealized, is a dynamic problem with more than one degree of freedom if upper is not rigid
- The simplest representation, which conforms to typical discrete inelastic models, has TWO aggregate, generalized coordinates
- A 1D solid slab and spring model with many members, and which fails, has a propensity for crush-up whether or not crush down occurs

Surprised? Not me. But maybe you disagree. I'll be back to disclaim, explain and support these notions, along with showing some results, as time permits. In the meantime, either yawn or rip me a new one, either one is fine.

The thread title comes from terms used by David B. Benson, Dr. G, and Heiwa. The placement is because the topic is not specific to the mechanics of any particular collapse.

About jolts. I just ran a 1D slab model, 110 story with 12 in the upper block. The upper block was made artificially stiff to force crush down first, with the connections set to the strength of the top lower connection plus a margin. The connections were such that the lowest could support 4x the imposed static load. The strength then scaled linearly up to the separation point, not terribly strong as you go up - it could go many times stronger towards the top before causing arrest.

Anyway, that's a little background. There isn't a stiffer situation to model, really, the upper block is a set of rigid masses coupled by very stiff springs. These springs break under minimal deflection. If there's going to be a jolt, it will be visible in this sort of system of discrete, axial strikes, where the only interaction is rigid body collisions, one per story. And they are visible.

This is the velocity of the 'roofline' over the first two seconds of the simulation, 500 samples per second:



Really obvious. Now how about the exact same data at 5 samples per second, first four seconds:



Not even a couple of flat spots, completely monotonic. Bearing in mind there's no video measurement error associated, this is about as much jolt as I'd expect to see when it is present, at that sample rate.

Not trying to pick on anyone, this is just something I've run across a lot and wanted to illlustrate. It doesn't matter if this is an accurate sim or whatever. It matters that the jolts are visible at one sample rate and not the other, exact same data.

If it wasn't obvious, an overlay shows what's going on:




Another run with a stronger structure produced a flat spot and a jolt, even at 5 samples per second:




But... that's with perfect recording accuracy. Let's be fair. Here's the displacement curve that goes with the velocity graph, only at 500 samples per second:




The task is to digitize the curve using whatever method, whatever resolution you like. It's got to be easier than digitizing an edge frame by frame in a grainy video, it's a graph after all. Then take differences to get the velocity. Trust me, there are beaucoup jolts in this curve. We can compare results to the 500 Hz data coming straight out of the engine and see how well you did.

Then do the same thing with a generated graph of a parabola, just to see how jittery a straight line can be.

OneWhiteEye --- I attempted to send the following as a PM, but it seems I failed.
=======
WTC 1 Horizontal Motion Thread

I'd like you to start a new thread to analyze the horizontal movements of the antenna mast dish. The idea is to look for jerks or jolts corresponding (I assume) to the various large sections of exterior wall which ripped away early on.

The goal is to establish the time of these occurances with the eventual idea of determining how much PE was released by the vertical motion and hence available to rip off those large sections.

Yes, I can do that.

David B. Benson wrote: The idea is to look for jerks or jolts corresponding (I assume) to the various large sections of exterior wall which ripped away early on.

I believe they are there.

The goal is to establish the time of these occurances with the eventual idea of determining how much PE was released by the vertical motion and hence available to rip off those large sections.

Sounds like a worthwhile endeavour.

Typical cusp example, hopefully much is self-explanatory.

- constant mass
- varying strength
- true 1D in 3D
- dynamics calculated using spheres
- rendered as slabs with height = diameter of spheres
- camera tracks with topmost member
- sequence ends at arrest

That covers most of what isn't obvious.

A comparison of displacement for two stretch values, 0.2 and 0.05, everything else the same. The collapses go to completion, being a mix of crush up and down until the top is completely crushed. Blue is the larger stretch value.



Nearly identical through the mixed phase, but the phase lasts longer for the smaller stretch, and this sets the difference between them until the end is reached, where once again the crush down (only) phase lasts longer since there's further to go.

This example also illustrates the characteristic appearance that these particular simulations produce when there is mixed crush direction. A little almost-parabola riding atop a larger almost-parabola.

Jumping into this without disclaimers is ill-advised. Nothing here is meant to represent a real structure, let alone any particular building. The simulator has not finished basic validation against knowns, nor has the energy consumed in breaking connections been fully profiled over the range of conditions of these examples. I'm using unit masses! Even if flawless, the inherent limitations of a discrete 1D analysis limit the application. There is some hope it will be illuminating, regardless.


The first case to examine closely is one in which a mix of crush up and down occur simultaneously in the beginning, but crush-up stops before the topmost connection is broken. Crush down goes to completion, but the final connection at the top does not break at the end.

The case of the top connection surviving has been encountered frequently. The reason is simple, and it goes to why my earlier simulations shown in the Crush Down Models thread were able to shatter the lower 90% without suffering damage. An upper block of only two slabs has a single slab bearing down on the connection, versus many slabs bearing down. If the lower block (Zone A) is being crushed down simultaneously, what takes place at the top is a low gravity crush up; that is, dynamics of the upper block (Zone C) is simply the solution of a crush up scenario in an accelerated frame wrt ground, i.e., a falling elevator. Zone C can rarely survive this pummeling even in the accelerated frame but, if any part of it can, it's the topmost part.

Back to the case at hand. First, an animation of the early drop, which ends at the point we see the topmost connection survive. Again, the camera tracks with the top, centered on a point below. This has proven to be the most useful view to capture the motion, though it might take a little time to feel comfortable. Each slab starts with two connections, shared above and below, except for the top and bottom slabs of Zones A and C. The number of remaining connections (2, 1, 0) to a given slab is reflected in the color of the slab.



The ratio of crush up/down is seen directly in the color bands. To achieve this particular mix, the entire upper block Zone C was set to the same connection strength throughout, to a value which is slightly higher than that of the top connection of the lower block Zone A. Later, this run will be compared to a model differing only by a very slight increase in the connection strength of Zone C, to make more of the upper block survive the initial drop.

Next, some graphs to get an idea of the dynamics for this simulation. Position of the very top over time:



Once again, the superimposed pair of curves are seen, corresponding to an initial phase of mixed crush direction followed by a crush down only phase. A kink is visible in the curve where crush up ceases, leaving the top two 'intact' slabs riding down on Zone B, the crush layer. The jolt can be seen in the velocity curve:



Can't miss that one.


The next graph follows the displacement over time of the top of each zone. Zone B, of course, doesn't exist until the first contact and breakage, so is defined as having zero position until it exists.



Notice how the complex curve of the top motion has been resolved into two distinct curves, the yellow and red. The crush up of Zone C, as well as its termination, seem to have little influence on the position of the other zone interfaces. Almost as if it could be treated as an independent problem. The difference between the blue and red curves depict the growing size of the crush layer, Zone B.


Less interesting, but still useful, is a graph showing the momentum possessed by Zones B and C as well as the total momentum (which includes 'stationary' Zone A, which can and does oscillate):




The distribution of energy over time shows the conversion from PE to KE with attendant losses from inelastic collisions and connection breakage. The green line is the accumulated dissipated energy:




Potential energy distribution amongst the zones, against the total available, mostly reflects the changing distribution of mass of each zone, with everything going towards B and a gradually reducing amount because of descent:




Kinetic energy, once crush up is done, is all Zone C for similar reasons:




Turning back to displacement, this is the displacement of the overall center of mass, which is quite smooth:




Next will be the same model tweaked ever so slightly to promote nearly exclusive crush down in the beginning.

In these models, the upper block connections have uniform strength greater than the top connection of lower block. If the upper block is strengthened by a small amount beyond that of the previous run, it fares better, staying mostly intact until the bottom. The end of crush down in that circumstance is where the crude simulation departs from anything a real structure might do - Zone B, the crush layer, compresses then rebounds up to meet the falling upper block. A 400+m lattice of highly inelastic balls stops acting like a building at this point; it's fair to ask whether such a simulation acts like a building at any point. If not, can the same question be asked of any similar discrete 1D model?

Incidentally, it seems the only way to tame this crazy explosion at the end (can you say pyroclastic clouds?) is to artificially weld the slabs together as they become part of Zone B, then fix B to the ground when it hits. I can only imagine it would play havoc with the solver, and I don't see a big advantage over studying crush up separately when riding a solid block down at a fraction of g, something that will be well-behaved. But I'll try welding, too, probably.

The end of the curves, as a result, should be taken with the appropriate grain of salt. The initial crush mix looks like this:



while the bounce-up at the end is:




Displacement of the very top is a single smooth curve, instead of being two descents superposed.



Detangling the displacements by zone shows a different picture than the previous example. The reflection in the Zone B (red) curve at the end is the bounce up.




The speed of the top, which is the roofline, shows a series of relatively small jolts; small considering Zone A+B is a group of semi-rigid slabs falling together. The most notable features are the bend in the beginning and the linearity of the two segments associated with the fall.



The first segment is essentially freefall, an unavoidable consequence of a simple slab model, the second segment is a shallower jaggy line composed of freefall descent peppered with jolts. Without high-resolution data, the line would appear straight with gaussian noise rather than unambiguously showing the customary sawtooth pattern of freefall punctated with jolts, as seen in this close up of the speed between seconds 4 and 6:



It's worth noting that this simulator gives an irregular pattern of jolts, simply because things are jostling around - as they would in the real world! The degree of similarity or difference between this and reality is one thing, the other things are the relationship between this and other models, their relationship to reality, and between different real scenarios. Is it more realistic to have uniformly regular collisions, as might happen in an analytical treatment which defines them to be so?

Comparing this example with the last reveals the gross differences in roofline motion but relatively minor differences in other parameters. The two collapses are virtually indistinguishable in the other measurable values. The previous example is labeled with suffix 0, this one with 1.

Displacement of the top:


Velocity of the top:


Displacement of center of mass:


Location of top of lower block (crush front):


Total kinetic energy:

If this style of simulation offers anything of value by way of comparison to physical progressive collapse, at least in these two examples, it is:

1) mixed crush direction should be distinguishable from exclusive crush down via roofline measurements of sufficient duration, by virtue of a faster descent

2) having mixed crush direction does little to alter the energetics of the overall problem, so ignoring it is OK (sometimes)

These observations run somewhat counter to my early intuition about the problem. I'd imagined simultaneous crush-up consumed more energy per unit time and would therefore leave less energy for motion. Wrong. The two 'reactions' occuring simultaneously liberate more PE per unit time by having the top drop correspondingly faster, roughly the sum of displacement due to crush down and up. Exactly so if defined as two decoupled processes, which seems a reasonable approximation.

Are these observations applicable to the real world? Is the real world WTC1, a 1D lattice of spheres - or both?

I think the lesson of top displacement under mixed crush applies generally to 1D models. At its core, the problem is approximately a loosely coupled pair of systems. The assumption of the prevailing theory, which is a continuum model, is that the upper block remains relatively undamaged until the end. I've read the part of B&L that addresses this (many times) and, while I understand the majority, I'm not in a position to check the validity of the work. I'll assume it to be correct and conclude that this is a significant difference with a discrete floor model - the upper block crushes up because it would if dropped onto a moving platform accelerating downward at a significant fraction of g, whereas in the case of continuous media this is not the case.

In a solid slab / stiff spring model like this, the initial collision is between two slabs which are coupled to other masses, NOT a collision between a rigid upper block and a single lower slab. The physics engine correctly applies the conservation of momentum to the two (mostly inelastic) bodies in contact. Then, based on resulting velocities of the bodies and forces transmitted through remaining connections, calculates the trajectories of the various pieces as independent bodies joined by stiff connections.

An upper block with the inertia of many members is not going to slow down much before the bottommost connection breaks and the resistance from Zone B is gone again for a time. Since the first collision is between two slabs, they move downward post-collision with half the speed of the upper block. The upper block catches up, breaking the lowest connection of Zone C, simple as that. The top will crush up right away unless the entire stucture is relatively weak and connection strength does not vary much going up. The other option is to make the top overly strong, as done here.

I wonder why the case of continuous media, discussed in B&L, differs so dramatically? Can an upper block that has experienced 10x maximum elastic deformation of its lowest story stand on its own at one-third gravity? Seems reasonable. How about when it's going to get smacked one more time before the ride smoothes out? Is not the extended treatment of B&L just a move of axiomatic rigidity one more story up?

One other thing falls out of this. Having the upper block be rigid as axiomatic does not greatly change the overall dynamics of a complete collapse, so seems justified from that standpoint. On the other hand, with such a simplified model that yet provides 110 degrees of freedom, violating this axiom (which is more realistic) sometimes leads to arrest.

You probably remember the Hambone comment about the conservation of momentum not applying to the slab collisions.

The nature of the floor connections, being indirectly connected to a very stationary earth, acts as an "outside force" during the collisions.

Conservation of momentum in a defined system with applied outside force during impact cannot be assumed.

What do you think about that?


The thread is great.

Yes, I do remember. Of course, it's true, but it also depends on how you look at it. If you include the earth in the system, then momentum is conserved, but that's not saying much of use. If a fast car hits a solid wall and comes to a dead stop, there's nothing to be gained by claiming momentum is conserved in the car-earth system. OK, momentum is not conserved, because of how we define our system.

If a car hits a second (stopped) car and the two, fused together, hit a wall, can we say momentum is conserved? In the first collision, yes, in the second collision and overall, no. What if the cars are in the process of fusing (i.e. the first collision is not over) when the second collision begins, as would be the case if the stopped car were parked an inch from the wall? Momentum is only conserved up until the imposition of an external constraint force.

But wait... the frictional force of the tires against the road is an external force. Put the parked car a mile away from the wall and the pair of fused cars will never reach it. Momentum was not conserved in the first place, after all. Now, what about having the parked car butted up against the wall, but have the wall just be a flimsy fence that breaks away? The head swims.



In this model, I say yes, momentum is conserved in slab collisions - more or less - and that's correct dynamics for this configuration. Only 'more or less' because there are connections that consume energy when they break, and that just happens to be in the same small region of time and space as the collision. These connections are then analogous to the breakaway wall in the example above. If the slabs were just suspended in space at equal distances, zero g and without connections, no question the collisions would be between individual slabs, not 'blocks', and momentum is conserved.

Add weak connections; not enough to hold the slabs against gravity. Momentum not conserved, technically, but in the limit of very weak connections, yes. Make them stronger, until strong enough to arrest immediately in gravity, no conservation at all.

If one takes the collision between slabs as instantaneous, then no distance is traveled during the collision. The bodies emerge from the collision together at the new, reduced velocity. Because there is no displacement, forces of constraint can do no work during the collision; connections are broken after subsequent displacement. With respect to individual bodies, momentum is conserved.

In actuality, this simulator uses a thin skin around bodies to allow some interpenetration and finite collision distances, as such is more realistic than a purely analytical computation based on an idealized assumption. Also, the connections break over a small but measurable distance at the time of collision so, practically speaking... momentum is not conserved.

Hahaha, isn't this grand?


This is what matters: while a connection survives, it transmits force to the next solid member, and so on. Once broken, in this discrete model, transmitted/resistive force due to connections goes to zero. Slab inertia remains.

An upper block may have X times the KE necessary to fail the first connection below, but it also has to accelerate the topmost lower slab to the upper block speed very quickly or the differential displacement will exceed the connection limit for the bottom of Zone C. We know there's enough energy to do so. The inelastic collision between the interface slabs only brings the fused pair of slabs to half the Zone C speed. Zone C can only be slowed by upward force applied through the surviving lowest connection, it is a mistake to take the collision as happening between all the slabs of Zone C and the one slab below. This same lowest connection applies downward force from Zone C motion to the pair of slabs in contact, and their inertia provides a reaction force.

If the newly formed Zone B, now consisting of one slab, and Zone C do not reach a common speed right away - by virtue of force transmitted through the single lowest connection of Zone C - that connection fails and Zone B then has two members. Zone C is still moving faster than Zone B. Crushing up.

How can the first slab of Zone B (fomerly top of Zone A) receive any more impulse from Zone C than the connection strength can provide before failure? Well, it can't. So it really doesn't matter if the upper block has a million times the KE to fail one connection, the limit to the force it can apply is the connection strength. Hence, simulations with varying connection strength (stronger at the bottom) will fail the lowest connection of Zone C under most circumstances, and crush up whether or not they continue to crush down.

This model, conceptually simple with its 'incompressible' inelastic slabs separated by empty space, has characteristics far from everyday experience with materials and structure. The simplicity allows exploration of collapse dynamics without too many factors muddying the waters, but is correspondingly limited in its output and application. It is arguably closer to reality than a continuous, uniform mass distribution, though, so its lessons are worth considering. This is as close to the blocks being rigid bodies as one can get without DEFINING them to be rigid, and they're just not rigid. Only the slabs are.

Major_Tom wrote:The thread is great.

Thanks. Excellent question.

The tortuous answer above contains one nugget. Momentum is largely conserved in these collisions, because it's an action that involves two slabs making contact, not a whole chain of slabs in either direction, acting as fully coherent rigid bodies. Whether energy is dissipated by breaking a connection at the same time over the same distance doesn't really matter, since it can be accumulated in the dissipated total as an intrinsic property of the body, with the bond, participating in the collision.

It matters very little that the lower block is fixed to the ground, other than keeping it in place. It does dissipate energy through loading and unloading, with these parameters, but a top slab impact is quite limited to the impulse it can transmit to the rest of the lower block before it breaks.

Conversely, it doesn't matter much that the upper block is a free body. When considering the upwardly directed impulse at collision, which acts to retard the motion of the upper block, it is again an impulse that must be transmitted through a connection which can fail. That impulse is transmitted only to the next slab, with mass, which also must be decelerated relative to the slab above it in order to transmit force through that connection, and so on. This is not a set of identical massless springs but masses between springs.

It's a situation like crumple zones on a car, and how they act in an accident. The farther away from the impact in a deformable body, the less acceleration experienced at the point. Impulse is spread out. It takes more distance and more time to slow down, at the expense of deformation and breakage.

I welcome your thoughts on the matter, Major_Tom, or anyone. Is there anything wrong with the simulation? If it says nothing about real buildings or the continuous/homogeneous models that have been developed, does it at least say something about discrete slab models? It tells me that, on the Crush Down Models thread, I want to keep the top block of my ceramic tile model limited to two stories. Has that come out of any other analysis?

This is a comparison between various mixes of crush up and down in a 110 story structure. There are differences needed to get the desired results but otherwise similar structure strengths. Two of the cases are from earlier.

- the predominant crush down by artificially stiff top, the one with rebound (red)
- the more natural mixed initial crush down/up, two curves superposed (yellow)

New cases are:

- exclusive crush down using only the top (two?) floor(s) (blue)
- exclusive crush up by dropping ENTIRE building at lowest floor (green)

Finally, freefall is plotted in dark red for comparison.

All displacement


Displacement 0-2sec


All speed


Comments:

Exclusive crush up is the closest to g. WTC7-ish. Without accurate hi-res measurements, there's little chance of distinguishing from freefall in the first 1.5 seconds. Exclusive crush down distinguishes itself early by 'skipping' like a stone on a pond, not a lot of mass driving from above and momentum transfer is a very big thing early on.

The velocity graph is more interesting. The yellow (mixed crush) matches the green (all crush up) until the upper block is getting small, then it starts to slow in a tail-off. When there's nothing left to crush up, it transitions abruptly over to the speed of the predominant crush down, which still has a mostly intact upper block. These two have the same acceleration as the exclusive crush down after four seconds, which is roughly constant (ignoring the bumps).

The constant average acceleration of the crush down portions is 30% of g.



That's very slow. And suspicious ([1 - stretch]*height/g = 0.3). What's wrong here? Is my simulator crap? Can't be too surprised about that.